Application of GARCH models in R – Part II ( APARCH)




If you read my previous post regards forecasting volatility using GARCH models you won’t have any trouble to forecast volatility by APARCH models.
APARCH model is the extended form of GARCH model that allowing to combine two effects in practical use (i.e. asymmetry in the impact of positive and negative lagged returns (i.e. leverage effects) and have flexible power in the volatility calculation).
The formula of APARCH is:

The APARCH model is one of the most complicated models in use. It allows for leverage effects when gamma is not zero and power effects when delta is not 2. If we impose both values to zero and 2 respectively we just get the GARCH model back.
You should note that in many case estimation will fail if the data sample is too short or exhibits structural breaks in volatility like what happened to SP500 index during the 2007-2009 crisis, or it will fail if the data sample size is too short.
If we restrict the asymmetry parameter ζ to a fixed value like zero, or set the power parameter δ = 2, estimation is feasible with a smaller sample.
Following code shows how you can use garchFit() function to estimate coefficients in different APARCH models.


# normal APARCH(1,1)
aparch_11=garchFit(formula= ~aparch(1,1), data=na.omit(Zero_mean_return)
         ,include.mean = FALSE,trace = F)
# fix delta at 2
aparch_11=garchFit(formula= ~aparch(1,1), data=na.omit(Zero_mean_return)
                   ,include.mean = FALSE,include.delta = F,delta = 2,trace = F).predict()
aparch_11@fit$coef
garchFit()

# normal APARCH(1,1)
aparch_11=garchFit(formula= ~aparch(1,1), data=na.omit(Zero_mean_return)
         ,include.mean = FALSE,trace = F)
# fix delta = 2
aparch_delta2_11=garchFit(formula= ~aparch(1,1), data=na.omit(Zero_mean_return)
                   ,include.mean = FALSE,include.delta = F,delta = 2,trace = F)
aparch_delta2_11@fit$coef

aparch_student_t_11 = garchFit(formula = ~aparch(1,1),data = na.omit(Zero_mean_return)
                               ,include.mean = FALSE, cond.dist = 'std',trace = F)
aparch_student_t_11@fit$coef

# Normal Aparch(2,2)
aparch_22 = garchFit(formula = ~aparch(2,2), data = na.omit(Zero_mean_return),
                     include.mean = FALSE ,trace=F)
aparch_22@fit$coef

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